Second-Order Memory and the Golden-Ratio Conjecture in Generator–Verifier Budget Allocation
Yunjing Chen · Zenodo (CERN European Organization for Nuclear Research) · 2026
Recursive self-improvement in AI systems is often described informally as a generator producing candidates and a verifier certifying them, with the verifier’s judgment propagating trust to further layers of a system. If the propagation is lossless and the layers are self-similar, a minimal two-layer linear model predicts that the equilibrium ratio between adjacent layers’ budgets is governed by the recurrence x_{n+1} = x_n + p x_{n-1},whose fixed ratio is 𝑟(𝜌) = 1/2 (1 + √(1 + 4p)) and which reduces to the golden ratio 𝜑 at 𝜌 = 1. This paper states that model precisely, derives its invariant axis and convergence rate, and then subjects its central quantitative prediction to two independent falsification tests grounded in an established engineering cost model: single-level speculative decoding and a three-level speculative-decoding cascade with recursive packing. In both tests the predicted ratio 𝑟(𝜌) is rejected: the median absolute deviation from prediction is on the order of one to several units of 𝑟, fewer than 2–10% of parameter settings fall within the predicted band, and the underlying cost recursion is shown analytically to be first-order (single-step memory, geometric contraction) rather than second-order (two-step memory, Fibonacci-type growth), which is the structural precondition the model requires. The paper’s contribution is therefore not an empirical claim that AI systems converge toward golden-ratio budget allocation — that hypothesis is rejected — but a falsifiable framework, a precise structural criterion (second-order memory) under which the golden-ratio limit is the only mathematically consistent outcome, and a demonstration that at least one major class of current inference-time systems does not meet that criterion. The result is offered as a boundary condition and a logical possibility for future layered-verification architectures, not as a description of present practice.